Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Lemma 5.1.2

The following conditions are equivalent for any unital -algebra

  1. is a finite C-star algebra.
  2. All isometries in are unitary.
  3. All projections in are finite projections.
  4. Every left-invertible element in is invertible.
  5. Every right-invertible element in is invertible.

Proof:
()
If is an isometry, then by direct calculation we know that is a projection. Hence and is a unitary.
()
If for some projection . Then by the definition of murray von neumann equivalence, there exists an isometry which connects and . Thus and .But in this algebra, every isometry is a unitary, so , this shows that is equivalent to a proper subprojection of itself.
()
Suppose that and are two projections in such that and . Let be a partial isometry in which implements the equivalence between and . and . Put and from our assumption, so we get that . We conclude that and , but since all isometries are unitary, necessarily, proving that every projection is equivalent to some proper sub projection of itself.
() If holds, then the unit is a finite projection.
() Parse through what it means.
() Trivial.
()
Suppose that holds, and let be a left-invertible element in . find such that . Recall that if are self-adjoint elements in a unital -algebra , then and if then for each . Use these facts to obtain the inequality This shows that and the spectrum of is therefore contained in the interval , which in particular implies tat is invertible. Put , and observe that therefore is invertible by (2), and hence is invertible.
Warnings:

  • A finite -algebra, unital or not, will always satisfy (3), that all projections are finite.
    The converse is NOT TRUE
  • A non-unital -algebra can be infinite and satisfy (3), have only finite projections.
  • Lastly, a finite C-star algebra does not have to be stably finite.
Table Of Contents
Lemma 5.1.2